Solution of Harmonic Problems with Weak Singularities Using Equilibrated Basis Functions in Finite Element Method
Authors
Abstract
In this paper, Equilibrated Singular Basis Functions (EqSBFs) are implemented in the framework of the Finite Element Method (FEM), which can approximately satisfy the harmonic PDE in homogeneous and heterogeneous media. EqSBFs are able to automatically reproduce the terms consistent with the singularity order in the vicinity of the singular point. The newly made bases are used as the complimentary enriching part along with the polynomial bases of the FEM to construct a new set of shape functions in the elements adjacent to the singular point. It will be shown that the use of the combined bases leads to the quality improvement of the solution function as well as its derivatives, especially in the vicinity of the singularity.
Keywords
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2. Möes, N., Dolbow, J., and Belytschko, T. “A Finite Element Method for Crack Growth without Rerneshing”, International Journal for Numerical Methods in Engineering, Vol. 46, pp.131- 150, 1999.
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8. Williams, M. L. “Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension”, Journal of Applied Mechanics, Vol. 19, pp. 526-528, 1952.
9. Li, Z. C., Chu, P. C., Young, L. J., and Lee, M. G. “Models of Corner and Crack Singularity of Linear Elastostatics and their Numerical Solutions”, Engineering Analysis with Boundary Elements, Vol. 34, pp. 533-548, 2010.
10. Li. Z. C., Lu, T. T., and Hu, H. Y. “The Collocation Trefftz Method for Bi-Harmonic Equations with Crack Singularities”, Engineering Analysis with Boundary Elements, Vol. 28, pp. 79-96, 2004.
11. Dolbow, J., Moes, N., and Belytschko, T. “Modeling Fracture in Mindlin-Reissner Plates with the Extended Finite Element Method”, International Journal of Solids and Structure, Vol. 37, pp. 7161-7183, 2000.
12. Ting, T. C. T., and Chou, S. C. “Edge Singularities in Anisotropic Composites”, International Journal of Solids and Structures, Vol. 17, pp. 1057-1068, 1981.
13. MantiČ, V., ParÍs, F., and CaÑas, J. “Stress Singularities in 2D Orthotropic Corners”, International Journal of Fracture, Vol. 83, pp. 67-90, 1997.
14. Wu. Z., and Liu. Y. “Analytical Solution for the Singular Stress Distribution due to V-Notch in an Orthotropic Plate Material”, Engineering Fracture Mechanics, Vol. 75, pp. 2367-2384, 2008.
15. Erdogan, F., and Wu, B. H. “Cracked Problems in FGM Layers Under Thermal Stresses”, Journl of Thermal Stresses, Vol. 19, pp. 237-265, 2007.
16. Bayesteh, H., and Mohammadi, S. “XFEM Fracture Analysis of Orthotropic Functionally Graded Materials”, Composites: Part B, Vol. 44, pp. 8-25, 2013.
17. Pathak, H. “Three-Dimensional Quasi-Static Fatigue Crack Growth Analysis in Functionally Graded Materials (FGMs) Using Coupled FE-XEFG Approach”, Theoretical and Applied Fracture Mechanics, Vol. 92, pp. 59-75, 2017.
18. Bhardwaj, G., Singh, I. V., Mishra, B. K., and Bui, T. Q. “Numerical Simulation of Functionally Graded Cracked Plates Using NURBS based XIGA under Different Loads and Boundary Conditions”, Composite Structures, Vol. 126, pp. 347-359, 2015.
19. Abdollahifar, A., Nami, M. R. and Saranjam, B. “Mixed-Mode Dynamic Fracture Analysis of FGM Plate by Mesh-Free Method”, Journal of Applied and Computational Sciences in Mechanics, Department of Engineering, Ferdowsi University of Mashhad, 2014. (In Farsi)
20. Soghrati, S. “Implementation of Smooth Fundamental Solutions in Solving Some Governing Differential Equations in Solid Mechanics”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2007. (In Farsi)
21. Mossaiby, F. “Solution of Solid Mechanics’ Problems in Bounded and Unbounded Domains using Semi-Analytic and Finite Element Methods”, Ph.D. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2010. (In Farsi)
22. Noormohammadi, N. “Solution of Solid Mechanics Problems Using Generalized Basis Functions”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2011. (In Farsi)
23. Noormohammadi, N. “Solution of Solid Mechanics Problems Using Equilibrated Basis Functions and Mesh-Free Methods”, Ph.D. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2015. (In Farsi)
24. Bateniparvar, O., Noormohammadi, N., and Boroomand, B. “Singular Functions for Heterogeneous Composites with Cracks and Notches; the Use of Equilibrated Singular Basis Functions”, Computers and Mathematics with Applications, Vol. 79, pp. 1461-1482, 2020.
25. Noormohammadi, N., and Boroomand, B. “A Fictitious Domain Method Using Equilibrated Basis Functions for Harmonic and Bi-Harmonic Problems in Physics”, Journal of Computational Physics, Vol. 272, pp.189-217, 2014.
26. Bateniparvar, O., “Solution of Problems with Weak Singularities in Heterogeneous Media Using Equilibrated Basis Functions and the Finite Element Method”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2019. (In Farsi)
27. Motz, H. “The Treatment of Singularities of Partial Differential Equations by Relaxation Methods”, Quarterly of Applied Mathematics, Vol. 4, pp. 371-377, 1947.
28. Szabo, B. A., and Babuska, I. “Computation of the Amplitude of Stress Singular Terms for Cracks and Reentrant Corners”, Washington University, Report WU/CCM, 1986.
29. Elliotis, M., Georgiou, G., and Xenophontos, C. “Solving Laplacian Problems with Boundary Singularities: a Comparison of a Singular Function Boundary Integral Method with the P/Hp Version of the Finite Element Method”, Applied Mathematics and Computation, Vol. 169, pp. 485-499, 2005.
30. Boroomand, B., and Noormohammadi, N. “Weakly Equilibrated Basis Functions for Elasticity Problems”, Engineering Analysis with Boundary Elements, Vol. 37, pp. 1712-1727, 2013.
2. Möes, N., Dolbow, J., and Belytschko, T. “A Finite Element Method for Crack Growth without Rerneshing”, International Journal for Numerical Methods in Engineering, Vol. 46, pp.131- 150, 1999.
3. Fleming, M., Chu, Y. A., Moran, B., and Belytschko, T. “Enriched Elemenr-Tree Galerkin Methods for Crack Tip Fields”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 483-504, 1997.
4. Hu, Y .Z., Li, Z. C., and Cheng, A. H. D., “Radial Basis Collocation Methods for Elliptic Boundary Value Problems”, Computers and Mathematics with Applications, Vol. 50, pp. 289-320, 2005.
5. Nasr-Esfahani, M. “Solution of Singular Problems Using Mesh-Less Local Exponential Basis Functions”, M.Sc. Thesis, Department of Civil Engineering, University of Isfahan, 2014. (In Farsi)
6. Mossaiby, F., Bazrpach, M., and Shojaei, A., “Extending the Method of Exponential Basis Functions to Problems with Singularities”, Engineering Computations, Vol. 32, pp. 406-423, 2015.
7. Brahtz, J. H. A., “Stress Distribution in a Re-Entrant Corner”, Transactions of ASME, Vol. 55, pp. 31-37, 1933.
8. Williams, M. L. “Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension”, Journal of Applied Mechanics, Vol. 19, pp. 526-528, 1952.
9. Li, Z. C., Chu, P. C., Young, L. J., and Lee, M. G. “Models of Corner and Crack Singularity of Linear Elastostatics and their Numerical Solutions”, Engineering Analysis with Boundary Elements, Vol. 34, pp. 533-548, 2010.
10. Li. Z. C., Lu, T. T., and Hu, H. Y. “The Collocation Trefftz Method for Bi-Harmonic Equations with Crack Singularities”, Engineering Analysis with Boundary Elements, Vol. 28, pp. 79-96, 2004.
11. Dolbow, J., Moes, N., and Belytschko, T. “Modeling Fracture in Mindlin-Reissner Plates with the Extended Finite Element Method”, International Journal of Solids and Structure, Vol. 37, pp. 7161-7183, 2000.
12. Ting, T. C. T., and Chou, S. C. “Edge Singularities in Anisotropic Composites”, International Journal of Solids and Structures, Vol. 17, pp. 1057-1068, 1981.
13. MantiČ, V., ParÍs, F., and CaÑas, J. “Stress Singularities in 2D Orthotropic Corners”, International Journal of Fracture, Vol. 83, pp. 67-90, 1997.
14. Wu. Z., and Liu. Y. “Analytical Solution for the Singular Stress Distribution due to V-Notch in an Orthotropic Plate Material”, Engineering Fracture Mechanics, Vol. 75, pp. 2367-2384, 2008.
15. Erdogan, F., and Wu, B. H. “Cracked Problems in FGM Layers Under Thermal Stresses”, Journl of Thermal Stresses, Vol. 19, pp. 237-265, 2007.
16. Bayesteh, H., and Mohammadi, S. “XFEM Fracture Analysis of Orthotropic Functionally Graded Materials”, Composites: Part B, Vol. 44, pp. 8-25, 2013.
17. Pathak, H. “Three-Dimensional Quasi-Static Fatigue Crack Growth Analysis in Functionally Graded Materials (FGMs) Using Coupled FE-XEFG Approach”, Theoretical and Applied Fracture Mechanics, Vol. 92, pp. 59-75, 2017.
18. Bhardwaj, G., Singh, I. V., Mishra, B. K., and Bui, T. Q. “Numerical Simulation of Functionally Graded Cracked Plates Using NURBS based XIGA under Different Loads and Boundary Conditions”, Composite Structures, Vol. 126, pp. 347-359, 2015.
19. Abdollahifar, A., Nami, M. R. and Saranjam, B. “Mixed-Mode Dynamic Fracture Analysis of FGM Plate by Mesh-Free Method”, Journal of Applied and Computational Sciences in Mechanics, Department of Engineering, Ferdowsi University of Mashhad, 2014. (In Farsi)
20. Soghrati, S. “Implementation of Smooth Fundamental Solutions in Solving Some Governing Differential Equations in Solid Mechanics”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2007. (In Farsi)
21. Mossaiby, F. “Solution of Solid Mechanics’ Problems in Bounded and Unbounded Domains using Semi-Analytic and Finite Element Methods”, Ph.D. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2010. (In Farsi)
22. Noormohammadi, N. “Solution of Solid Mechanics Problems Using Generalized Basis Functions”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2011. (In Farsi)
23. Noormohammadi, N. “Solution of Solid Mechanics Problems Using Equilibrated Basis Functions and Mesh-Free Methods”, Ph.D. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2015. (In Farsi)
24. Bateniparvar, O., Noormohammadi, N., and Boroomand, B. “Singular Functions for Heterogeneous Composites with Cracks and Notches; the Use of Equilibrated Singular Basis Functions”, Computers and Mathematics with Applications, Vol. 79, pp. 1461-1482, 2020.
25. Noormohammadi, N., and Boroomand, B. “A Fictitious Domain Method Using Equilibrated Basis Functions for Harmonic and Bi-Harmonic Problems in Physics”, Journal of Computational Physics, Vol. 272, pp.189-217, 2014.
26. Bateniparvar, O., “Solution of Problems with Weak Singularities in Heterogeneous Media Using Equilibrated Basis Functions and the Finite Element Method”, M.Sc. Thesis, Department of Civil Engineering, Isfahan University of Technology, 2019. (In Farsi)
27. Motz, H. “The Treatment of Singularities of Partial Differential Equations by Relaxation Methods”, Quarterly of Applied Mathematics, Vol. 4, pp. 371-377, 1947.
28. Szabo, B. A., and Babuska, I. “Computation of the Amplitude of Stress Singular Terms for Cracks and Reentrant Corners”, Washington University, Report WU/CCM, 1986.
29. Elliotis, M., Georgiou, G., and Xenophontos, C. “Solving Laplacian Problems with Boundary Singularities: a Comparison of a Singular Function Boundary Integral Method with the P/Hp Version of the Finite Element Method”, Applied Mathematics and Computation, Vol. 169, pp. 485-499, 2005.
30. Boroomand, B., and Noormohammadi, N. “Weakly Equilibrated Basis Functions for Elasticity Problems”, Engineering Analysis with Boundary Elements, Vol. 37, pp. 1712-1727, 2013.
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